A single scale smooth Alpert trilinear characterization of the Fourier extension conjecture on the paraboloid in three dimensions

TL;DR

This paper establishes an equivalence between the Fourier extension conjecture on the paraboloid in three dimensions and a new local single scale smooth Alpert trilinear inequality, improving previous multiscale results.

Contribution

It introduces a novel local single scale smooth Alpert trilinear inequality that refines earlier multiscale inequalities, linking it directly to the Fourier extension conjecture.

Findings

Equivalence between Fourier extension conjecture and a local single scale inequality

Development of an improved Alpert trilinear inequality

Reduction of a multiscale inequality to a single scale setting

Abstract

We show that the Fourier extension conjecture on the paraboloid in three dimensions is equivalent to a local single scale smooth Alpert trilinear inequality, which is an improvement of an analogous multiscale trilinear inequality in arXiv:2506.03992.